An Adiaba t ic Theorein f o r Motions Which Exh ib i t I n v a r i a n t Phase Curves

Kei th R. Symon, Wayne Un ive r s i t y , August 4, 1954

We cons ide r a p a r t i c l e w i t h one degree of freedom moving

according t o a Hamiltonian f u n c t i o n H(p ,q , ,A, t ) , depending on a parameter , and which may a l s o depend on t h e time. We

assume t h a t f o r each f i x e d value of h of i n t e r e s t , t h e motion i s s t a b l e and e x h i b i t s i n v a r i a n t phase curves , t h a t i s , curves

which a r e e i t h e r con t inuous ly o r p e r i o d i c a l l y transformed i n t o

themselves by t h e t ransformat ion gene ra t ed by t h e tIamiltonian

- equa t ions of motion. Th i s imp l i e s t h a t t h e r e i s a c o n s t a n t of -

t he motion c & ( p , q , ) \ , t ) , which i s e i t h e r independent of t o r p e r i o d i c i n t, t h e i n v c r i a n t curves being g iven by t h e equa t ion

d ( p , q , ), , t ) = d- = a cons t an t . (1)

When t h e Hamiltonian i s independent of t, the i n v a r i a n t curves

arc! s imply t h e c lo sed o r b i t s on t h e phase plane. When t h e

Hainiltonian i s p e r i o d i c i n t , t h e r e i s evidence, a t l e a s t i n

many cases , f o r t h e e x i s t e n c e of i n v a r i a n t s of t h e form (1) w i t h t h e same p e r i o d i n t. We w i l l f u r t h e r assume t h a t t h e r e

i s on ly one s e t of i n v a r i a n t curves , t h a t i s , t h a t a l l

c o n s t a n t s of t h e inotion a r e f u n c t i o n s o f d . This seems t o be t r u e except i n t h e case o f t h e l i n e a r a l t e r n a t i n g g r a d i e n t

-

- equa t ions whe:~ tile b e t a t r o n per iod i s a r a t i o n a l m u l t i p l e of

t h e s e c t o r per iod; i n t h e l a t t e r ca se i t i s probable t h a t t h e

a d i a b a t i c theorem can f a i l ,

The theorem t o be :>roved i s t h a t under t h e above

t issunptions, i f t h c parameter ,,A! i s changed s u f f i c i e n t l y slowly ( i n comparison wi th t h e per io6 of o s c i l l a t i o n o r t h e b e t a t r o n p e r i o d ) , a s e t of p a r t i c l e s which l i e i n i t i a l l y on a n i n v a r i a n t curve w i l l con t inue t o l i e on so-ie i n v a r i a n t

curve. L i o u v i l l e f s theorem s t a t e s t h a t a s e t of p a r t i c l e s

which l i e on any c l o s e d curve ( n o t n e c e s s a r i l y i n v a r i a n t ) w i l l a f t e r any change i n parameters l i e on n c lo sed curve haviilg

t h e saqe a rea . An immediate c o r o l l a r y of t h e s e two theorems

i s t h a t f o r s u l ' f i c i e n t l y slow c l zan~es i n t h e p a r a n e t s r ,,A, , t h e a r e a of t:.le i n v a r i a n t curve on wklicil a p a r t i c l e i s l o c a t e d

w i l l re-,nain cons t an t . h'heLn i;he ~ i ~ n i l t o n i a n i s independent of -

P t , t h i s reduces t o t h e u s u a l state7:ient or' t h e a d i a b a t i c

i nva r i ance of t h e a r e a 01' trle o r b i t i n the phase plane.

I n o r d e r t o nrove t l ~ e theorein, we show t h a t t h e change

i n i when ,)\ i s v a r i e d s lowly, i s a t'unc.tion ozlly oi' t h e i n i t i a l va lue of a , and heilca t h a t p a r t i c l e s hav ing i n i t i a l l y

t h e sane va lue of iJ w i l l con t inue t o have t h o saj'lc va lues a t . The t ime r a t e oi' change i n i s

. .

d, - -- A* + --- C ) & ; + d_nL. +-d-& ,!, < d l ' d ? 6 d * A

The sum of tilo f i r s t tiweit tor . , !~ i s zero , z ince ol i s a c o n s t a h t

of t h e motion f o r constanL A . Suppose ilow t h a t a t t = tl, A = anci c i u r i l ~ : tile t i n e t2 - tl changes t o 's 1'

C

- h2 = + A )\ , Me w i l l talre & h s ' . ~ f f i c i e n t l y srnall

so t h a t t h e c h a r a c t e r of t h e motion i s n o t app rec i ab ly

d i f f e r e n t f o r va lues of A between h and h 2, and we w i l l s e t

where j( f o r cbnveniance i s t o be h e l d c o n s t a n t d u r i n g t h i s t ime t 2 - ti . The change i n &, i s

Since we a r e t a k i n g A t o be c o n s t a n t , we can w r i t e t h i s i n t h e form

The r i g h t s i d e i s t h e time average of 3 ~ ' / 3 ~ dur ing t h e time t 2 - tl. If we assu-ne t h a t we can c a l c u l a t e t h i s average f o r

a f i x e d A between A1 and A 2 ( A A s u f f i c i e n t l y s m a l l ) , .

t hen i f t he t i n e t 2 - tl i s made su f ; i c i en t ly l ong ( s m a l l ) , t h e e rgodic theorem s t a t e s t h a t t he r i g h t s i d e of ( 5 ) may be ..lade t o approach a cons t an t o f t i l e no t ion , and hence i s a

f u n c t i o n of D( :

-4- For a d i a b a t i c v a r i a t i o n s i n ), , t h e chainbe i n d i s t h e r e f o r e governed by t h e d i f f e r e n t i a l equa t ion

whose s o l u t i o n w i l l depend o n l y on t h e i n i t i a l v a l u e of CX .

The e s s e n t i a l p o i n t i n t h e a p p l i c a t i o n of t h e c rgod ic

theorem i s t h a t i f A changes s u f f i c i e n t l y slowly, t h e p a r t i c l e w i l l have t i n e t o pas s aany t imes t h r o u ~ h a l l p a r t s of t h e phase

space corresponding t o i t s va lue of' , and h e m e t h e time

average of dd \ w i l l n o t depend on where the p a r t i c l e i s a t any p a r t i c u l a r time. I t i s t h e r e f o r e n e c e s s a r y t h a t t h e

A t ime r e q u i r e d f o r a s i g n i f i c a n t change i n :olust be xany -

o s c i l l a t i o n per iods . How many w i l l depend upon t h e d e t a i l s of

t h e motion and t h e n a t u r e of t h e f u n c t i o n / , If a p o i n t on t h e i n v a r i a n t curve i s alrnost f i x e d , o r allilost

p e r i o d i c w i th low p e r i o d i c i t y , i t may be neces sa ry t o go through

very many o s c i l l a t i o n s i n o r d e r t o sample t h e e n t i r e i n v a r i a n t

curve i n t h e t ime average of >oc/JA . I t i s c l e a r t h a t s i n c e t h e o s c i l l a t i o n p e r i o d becomes i n f i n i t e on a s e p a r a t r i x

s e p a r a t i n g two d i f f e r e n t t ypes of motion i n t h e phase plane,

the t r a n s i t i o n from one type of :notion t o ano the r can lever be

a d i a b a t i c .

A s a s imple a p p l i c a t i o n of t h e above theorem ( sugges t ed by F. T. Cole) , we coilslder a sil-nple harmonic o s c i l l a t i o n

C r s i th EL ;~;lr'aonic drivinr,. term:

2 + lJ :~= ,qcm w t . The Hamiltonian i s

The s o l u t i o n , i f t he parameter A is cons t an t , i s

where C , e a r e a r b i t r a r y cons tan ts . The f i r s t t e r m have t h e

same p o r i ~ d i c i t g a s the IIarniltonian. The t r a n s i e n t terms

r e p r e s e n t a n o t i o n around an e l l i p s e w i t h f requency ~ 3 , ; t h e

c e n t e r of t he e l l i p se , r e p r e s e n t e d by t h e s t eady s t a t e terins,

moves r i i th f requency LJ , and hence t h e e l l i p s e r e t u r n s

p e r i o d i c a l l y t o t h e same p o s i t i o n . The t r a n s i e n t e l l i p s e :

-6- i s an i n v a r i a n t curve. According t o t h e aciiaba-tic theorem

proved above, t h e a r e a o l t h i s curve r e a n i n s cons t an t f o r slow

c h a n ~ e s i n the paraineter A , t h a t i.s, t h e t r a n s i e n t ampli tude

remains constant . We can v e r i f y thal; t i l e conc lus ion i s c o r r e c t

i n t h i s ca se by n o t i n g t h a t i f we s e t

A = a t , (13)

then Eq. ( 8 ) has t h e exac t s o l u t i o n

wher-e C, 8 a r e a r b i t r a r y comtants, If - a i s van i sh ing ly smal l ,

tho l.ast t e r n s a r e n e g l i s i b l e , and t h e motion c o n s i s t s of t h e

s teady s t a t e p l u s a t r a n s i e n t of cons t an t ampli tude, i n agreeaient

w i t h t h e a d i a b a t i c theorem.